Equivalence Relations and Quotient Sets - Verifying a Claim

[matheater]

I have a question...
"Is the quotient set of a set S relative to a equivalence relation on S a subset of S?"
I suppose "no",since the each member of the quotient set is a subset of S and consequently it is a subset of the power set of S,but I have e book saying that "yes",I am a bit confused,can anyone ensure me?

 

[HallsofIvy]

No, the quotient set of set S, relative to some equivalence relation on S, is not a subset of S. It is the set of all equivalence classes defined by the relation and so, as you say, a subset of the power set of S. We could, by choosing one "representative" of each equivalence class, "identify" the quotient set with a subset of S but that can be done in many different ways depending on the choices of "representative".

For example, if S= N, the natural numbers, and the relation is xRy if and only if x-y is a multiple of 3, then the quotient set is the set containing: the multiples of 3, the set of numbers of the form 3n+1, and the set of numbers of the form 3n+2. We can, and often do identify those with {0, 1, 2}. But we could as easily identify them with {3, 4, 5}, etc.

 

[matheater]

Thank u very much,I am complete agree with u.